Definition 6.12 ([1]}) If \(R\)  is Noetherian (not necessarily local, or of prime characteristic) and if \(M,N \in \operatorname{Mod}(R)\) , then the grade of the pair \((M,N)\)  is defined as:
\(\operatorname{\textnormal {grade}}_R(M,N)=\inf \lbrace i\mid \operatorname{Ext}^i_R(M,N)\ne 0\rbrace .\) 

